The problem you're asking is about a $10 \times 10$ grid, where each cell contains a digit from 0 to 9. Each digit appears exactly 10 times in the grid. We are asked to prove that there is at least one row or one column that contains at least 4 distinct digits.

Let's break down the problem:

Known Information:

1. The grid is $10 \times 10$ in size.
2. Each digit (from 0 to 9) appears exactly 10 times in the grid.
3. We need to prove that there is a row or a column in the grid that contains at least 4 distinct digits.

Approach:

• Row and Column Sum: In total, there are 100 cells in the grid, and since each digit from 0 to 9 appears 10 times, the grid is fully filled with these digits.
• Pigeonhole Principle: This principle is often useful in problems like these. If we assume for contradiction that every row and every column contains fewer than 4 distinct digits, each row and column would be composed of only 3 or fewer distinct digits. But this would cause a contradiction because we are trying to distribute 10 distinct digits across the entire grid.

Steps to Prove:

1. Assume the contrary: Suppose that every row and column has fewer than 4 distinct digits. This means each row and each column would contain at most 3 different digits.

2. Count the total occurrences: There are 10 digits, and each appears exactly 10 times. So there are 100 cells to fill.

3. Contradiction: If each row has at most 3 distinct digits, the total number of distinct digit combinations in any 10x10 grid would not suffice to accommodate 10 distinct digits in a way that ensures each digit appears 10 times. This leads to a contradiction, thus proving that there must be at least one row or one column with at least 4 distinct digits.

Would you like a more detailed breakdown of this proof? Feel free to ask for clarifications.

Here are five related questions to expand on this:

1. How does the pigeonhole principle apply to similar grid-based problems?
2. What happens if the grid size is changed to $n \times n$ with different conditions?
3. How can you extend this proof to larger grids or higher distinct digit counts?
4. Could there be cases where a row has fewer than 4 distinct digits but a column compensates?
5. How does symmetry in the grid affect the distribution of digits?

Tip: When tackling problems involving distribution and distinct counts, the pigeonhole principle is often a useful tool for proving existence statements like this one.